ar X iv : q ua nt - p h / 05 04 18 9 v 2 2 8 Se p 20 05 OPERATOR QUANTUM ERROR CORRECTION

This paper is an expanded and more detailed version of the work [1] in which the Operator Quantum Error Correction protocol was introduced. This is a new scheme for the error correction of quantum operations that incorporates the known techniques — i.e. the standard error correction model, the method of decoherence-free subspaces, and the noiseless subsystem method— as special cases, and relies on a generalized mathematical framework for noiseless subsystems that applies to arbitrary quantum operations. We also discuss a number of examples and introduce the notion of “unitarily noiseless subsystems”. A unified and generalized approach to quantum error correction, called Operator Quantum Error Correction (OQEC), was recently introduced in [1]. This model unifies all of the known techniques for the error correction of quantum operations – i.e. the standard model [2, 3, 4, 5], the method of decoherence-free subspaces [6, 7, 8, 9] and the noiseless subsystem method [10, 11, 12] – under a single umbrella. An important new framework introduced as part of this scheme opens up the possibility of studying noiseless subsystems for arbitrary quantum operations. This paper is an expanded and more detailed version of the work [1]. We provide complete details for proofs sketched there, and in some cases we present an alternative “operator” approach that leads to new information. In particular, we show that correction of the general codes introduced in [1] is equivalent to correction of certain operator algebras, and we use this to give a new proof for the main testable conditions in this scheme. In addition, we discuss a number of examples throughout the paper, and introduce the notion of “unitarily noiseless subsystems” (UNS) as a relaxation of the requirement in the noiseless subsystem formalism for immunity to errors. We also connect this work with aspects of more recent OQEC related efforts. In particular, we show that the fundamental formula in the formulation of the “Quantum Computer Condition” recently introduced in [13] is captured as a special case of the UNS framework.